Journal of Statistical Software

Transcription

1 JSS Journal of Statistical Software January 2005, Volume 12, Issue 6. spatstat: An R Package for Analyzing Spatial Point Patterns Adrian Baddeley University of Western Australia Rolf Turner University of New Brunswick Abstract spatstat is a package for analyzing spatial point pattern data. Its functionality includes exploratory data analysis, model-fitting, and simulation. It is designed to handle realistic datasets, including inhomogeneous point patterns, spatial sampling regions of arbitrary shape, extra covariate data, and marks attached to the points of the point pattern. A unique feature of spatstat is its generic algorithm for fitting point process models to point pattern data. The interface to this algorithm is a function ppm that is strongly analogous to lm and glm. This paper is a general description of spatstat and an introduction for new users. Keywords: conditional intensity, edge corrections, exploratory data analysis, generalised linear models, inhomogeneous point patterns, marked point patterns, maximum pseudolikelihood, spatial clustering. 1. Introduction spatstat is one of several packages in the R language for analysing point patterns in two dimensions. 1 This paper is a general description of spatstat and may serve as an introduction for new users. Subsequent papers will cover advanced use of the package Baddeley and Turner (2005b) and explain its design and implementation Baddeley and Turner (2005a). A simple example of a point pattern dataset is shown in Figure 1. The points represent the locations of seedlings and saplings of the Californian giant redwood. Point pattern data may be much more complicated than Figure 1 suggests. The spatial sampling region in which the points were recorded may have arbitrary irregular shape, instead of being a rectangle as in Figure 1. The points may carry additional data (marks). For example, we may have recorded the height or the species name of each tree. There may be 1 Alternatives include splancs Rowlingson and Diggle (1993); Bivand (2001), spatial Ripley (2001); Venables and Ripley (1997), ptproc Peng (2003) and SSLib Harte (2003).

2 2 spatstat: An R Package for Analyzing Spatial Point Patterns Figure 1: The classic Redwoods dataset Ripley (1977) available in spatstat as redwood. additional covariate data which must be incorporated in the analysis. The spatstat package is designed to handle all these complications. Figure 2 shows an example of a dataset which can be handled by spatstat; it consists of points of two types (plotted as two different symbols) and is observed within an irregular sampling region which has a hole in it. The label or mark attached to each point may be a categorical variable, as in the Figure, or a continuous variable. See also Figures 6 9. Figure 2: Artificial example demonstrating the complexity of datasets which spatstat can handle. Point patterns analysed in spatstat may also be spatially inhomogeneous, and may exhibit dependence on covariates. The package can deal with a variety of covariate data structures. It will fit point process models which depend on the covariates in a general way, and can also simulate such models. 2. Goals Our main reasons for writing spatstat were to: Implement functionality. The research literature on spatial statistics provides a large body of techniques for analysing spatial point patterns (e.g. Bartlett (1975); Cliff and Ord (1981); Cressie (1991); Diggle (2003); van Lieshout (2000); Matérn (1986); Møller and Waagepetersen (2003); Moore (2001); Ripley (1981, 1988); Stoyan, Kendall, and Mecke (1995); Stoyan and Stoyan (1995); Upton and Fingleton (1985)). However, only a small fraction of these techniques have been implemented in software for general use.

3 Journal of Statistical Software 3 Handle real datasets. New techniques published in the literature are often demonstrated only on a tame example dataset, using a rudimentary proof-of-concept implementation. Such software is typically designed only for rectangular windows; the techniques themselves may assume that the point pattern is spatially homogeneous; and auxiliary information (such as covariate data) is often ignored. For example, the classical redwood dataset of Figure 1 is a subset extracted by Ripley (1976, 1981) from a larger dataset of Strauss (1975) which is shown in Figure 3. The full dataset exhibits completely different spatial patterns on either side of the diagonal line shown on the plot. The diagonal line is a simple example of covariate data. As far as we are aware, the full dataset has never been subjected to comprehensive analysis. Strauss s redwood data Region II Ripley s subset Region I Figure 3: The full redwood dataset of Strauss (1975). The square in the bottom left corner shows the boundaries of the subset extracted by Ripley (1977) as the classical redwood dataset. Similarly, Figure 4 shows the ant nest data of Harkness and Isham (1983). The full dataset records the locations of nests of two species of ants, observed in an irregular convex polygonal boundary, together with annotations showing a foot track through the region, and the boundary between field and scrub areas inside the region. Rectangular subsets of the data (marked A and B on the Figure) were analysed in Harkness and Isham (1983); Isham (1984); Takacs and Fiksel (1986); Högmander and Särkkä (1999); Baddeley and Turner (2000) and (Särkkä 1993, section 5.3). Again, as far as we are aware, the full dataset has never been subjected to detailed analysis inside the correct window. Fit realistic models to data. In applications, the statistical analysis of spatial point patterns is conducted almost exclusively using exploratory summary statistics such as the K function Cliff and Ord (1981); Cressie (1991); Diggle (2003); Møller and Waagepetersen (2003); Ripley (1988); Stoyan et al. (1995); Stoyan and Stoyan (1995); Upton and Fin-

4 4 spatstat: An R Package for Analyzing Spatial Point Patterns gleton (1985). An important goal of spatstat is to fit parametric models to spatial point pattern data. Although methods for fitting point process models have been available since the 1970 s Besag (1975); Diggle (2003); Ogata and Tanemura (1981, 1984); Møller and Waagepetersen (2003); Ripley (1981, 1988), most of these methods were very specific to the chosen model, and there were no software implementations of sufficient generality to fit realistic models to a real dataset. Recently we described an algorithm for fitting point process models of very general form Baddeley and Turner (2000). Our implementation of this algorithm has grown into the package spatstat. ants A B Figure 4: Harkness-Isham ant nests data. Map of the locations of nests of two species of ants, Messor wasmanni ( ) and Cataglyphis bicolor ( ) in an irregular region 425 feet in diameter. Data kindly supplied by Professors R.D. Harkness and V. Isham. spatstat supports the following activities. 3. Capabilities Creation, manipulation and plotting of point patterns: a point pattern dataset can easily be created, plotted, inspected, and transformed. Subsets of the pattern can easily be extracted (e.g. to thin the points or trim the window). Marks can readily be added or removed from a point pattern. Many geometrical transformations, operations and measurements are implemented. Exploratory data analysis: standard empirical summaries of the data, such as the average intensity, the K function Ripley (1977) and the kernel-smoothed intensity map, can easily

5 Journal of Statistical Software 5 be generated and displayed. Many other empirical statistics are implemented in the package, including the empty space function F, nearest neighbour distance function G, pair correlation function g, inhomogeneous K function Baddeley, Møller, and Waagepetersen (2000), second moment measure, Bartlett spectrum, cross-k function, cross-g function, J-function, and mark correlation function. Our aim is eventually to implement the vast majority of the statistical techniques described in the spatial statistics literature (e.g. Diggle (2003); Stoyan and Stoyan (1995)). Parametric model-fitting: a key feature of spatstat is its generic algorithm for fitting point process models to data. The point process models to be fitted may be quite general Gibbs/Markov models; they may include inhomogeneous spatial trend, dependence on covariates, and interpoint interactions of any order (i.e. not restricted to pairwise interactions). Models are specified using a formula in the R language, and are fitted using a single function ppm analogous to glm and gam. A fitted model can be printed, plotted, predicted, updated, and simulated. Capabilities for residual analysis and model diagnostics will be added in version 1.6. Simulation of point process models: spatstat can generate simulated realisations of a wide variety of stochastic point processes. Some process parameters (intensity function, cluster distribution) may be arbitrary, user-supplied functions in the R language. Markov point process models of a very general kind (including arbitrary spatial inhomogeneity and user-supplied interaction potential) are simulated using a fast Fortran implementation of the Metropolis-Hastings algorithm. Fitted model objects obtained from the model-fitting algorithm can be simulated directly by Metropolis-Hastings. 4. Demonstration A few examples of spatstat s capabilities are shown in the following transcript of an R session. A more extensive demonstration can be seen by installing the package and typing demo(spatstat). R> library(spatstat) R> data(cells) R> cells planar point pattern: 42 points window: rectangle = [0,1] x [0,1] R> plot(cells) R> plot(ksmooth.ppp(cells)) R> plot(kest(cells)) These commands performed some exploratory analysis of the dataset cells. The last two lines displayed a kernel-smoothed estimate of the intensity, and an estimate of the K function.

6 6 spatstat: An R Package for Analyzing Spatial Point Patterns R> fit <- ppm(cells, ~1, Strauss(r=0.1)) R> fit Stationary Strauss process beta interaction distance: 0.1 Fitted interaction parameter gamma: [1] R> Xsim <- rmh(fit) R> plot(xsim) This code fits a Strauss point process model to the cells data. The object fit is a fitted point process model. The code prints a summary of the fitted model, then simulates a realisation from this fitted model. R> data(demopat) R> plot(demopat, box=false) R> plot(split(demopat)) R> plot(alltypes(demopat, "K")) This code analyzes the point pattern shown in Figure 2 which consists of points of two different types. The split command separates the dataset into two point patterns according to their types, which are then plotted separately. The alltypes command computes the bivariate ( cross ) K function K ij (r) for each pair of types i, j and plots them as a 2 2 array of graphs. R> pfit <- ppm(demopat, ~marks + polynom(x,y,2), Poisson()) R> plot(pfit) The call to ppm fits a non-stationary Poisson point process to the data in Figure 2. The logarithm of the intensity function of the Poisson process is described by the R formula ~marks + polynom(x,y,2) which represents a log-quadratic function of the cartesian coordinates, multiplied by a constant factor depending on the type of point. The last line plots the fitted intensity function as a perspective view of a surface. 5. Data types The basic data types in spatstat are Point Patterns, Windows, and Pixel Images. A point pattern is a dataset recording the spatial locations of all events or individuals observed in a certain region. A window is a region in two-dimensional space. It usually represents the study area. A pixel image is an array of brightness values for each grid point in a rectangular grid inside a certain region. It may contain covariate data (such as a satellite image) or it may be the result of calculations (such as kernel smoothing).

7 Journal of Statistical Software 7 Figure 5: A point pattern, a window, and a pixel image. spatstat uses the object-oriented features of R ( classes and methods ) to make it easy to manipulate, analyse, and plot these datasets. Note that there is no predetermined format for covariate data. Indeed that would be unnecessarily limiting, as there are many different kinds and formats of covariate information that might be needed. Our modelling and simulation code accepts covariate data in various formats Point patterns A point pattern is represented in spatstat by an object of the class "ppp". A dataset in this format contains the coordinates of the points, optional mark values attached to the points, and a description of the spatial region or window in which the pattern was observed. To create a point pattern (class "ppp") object we may create one from raw data using the function ppp, convert data from other formats (including other packages) using as.ppp, read data from a file using scanpp, manipulate existing point pattern objects using a variety of tools, or generate a random pattern using one of the simulation routines. For example, to create a pattern of random points inside the rectangle [0, 10] [0, 3], R> x <- runif(20, max=10) R> y <- runif(20, max=3) R> u <- ppp(x, y, c(0,10), c(0,3)) x The Venables and Ripley spatial library, which is part of the standard distribution of R, supplies a dataset pines. To convert this into our format, R> library(spatial) R> pines <- ppinit("pines.dat") R> library(spatstat) R> pines <- as.ppp(pines) A point pattern must have a window Note especially that, when you create a new point pattern object, you need to specify the spatial region or window in which the pattern was observed. We believe that the observation window is an integral part of the point pattern. A point pattern dataset consists of knowledge about where points were not observed, as well as the

8 8 spatstat: An R Package for Analyzing Spatial Point Patterns locations where they were observed. Even something as simple as estimating the intensity of the pattern depends on the window of observation. It would be wrong, or at least different, to analyze a point pattern dataset by guessing the appropriate window (e.g. by computing the convex hull of the points). An analogy may be drawn with the difference between sequential experiments and experiments in which the sample size is fixed a priori. For situations where the window is really unknown, spatstat provides the function ripras to compute the Ripley-Rasson estimator of the window, given only the point locations Ripley and Rasson (1977). Marked point patterns Each point in a spatial point pattern may carry additional information called a mark. For example, a pattern of points which are classified into two or more different types (on/off, case/control, species, colour, etc) may be regarded as a pattern of marked points, where the mark attached to each point indicates which type it is. Data recording the locations and heights of trees in a forest can be regarded as a marked point pattern where the mark attached to a tree s location is the height of the tree. In our current implementation, the mark attached to each point must be a single value (which may be numeric, character, complex, logical, or factor). Many of the functions in spatstat for marked point patterns require that the mark attached to each point be either a continuous variate or real number. An example is the Longleaf Pines dataset (longleaf) in which each tree is marked with its diameter at breast height. The marks component must be a numeric vector such that marks[i] is the mark value associated with the ith point. We say the point pattern has continuous marks. a categorical variate. An example is the Amacrine Cells dataset (amacrine) in which each cell is identified as either on or off. Such point patterns may be regarded as consisting of points of different types. The marks component must be a factor such that marks[i] is the label or type of the ith point. We call this a multitype point pattern and the levels of the factor are the possible types. See Figures 6 7. Note that, in some other packages, a point pattern dataset consisting of points of two different types (A and B say) is represented by two datasets, one representing the points of type A and another containing the points of type B. In spatstat we take a different approach, in which all the points are collected together in one point pattern, and the points are then labelled by the type to which they belong. An advantage of this approach is that it is easy to deal with multitype point patterns with more than 2 types. For example the classic Lansing Woods dataset represents the positions of trees of 6 different species. This is available in spatstat as a single dataset, a marked point pattern, with the marks having 6 levels. Standard datasets Some standard point pattern datasets are supplied with the package. They are summarised in Table 1.

10 10 spatstat: An R Package for Analyzing Spatial Point Patterns longleaf Figure 6: Point pattern with continuous marks (tree diameter). The Longleaf Pines dataset Platt et al. (1988); Rathbun and Cressie (1994), available as longleaf. Figure 7: Point pattern with categorical marks (cell type). Hughes amacrine cell dataset Diggle (1986), available as amacrine Windows An object of the class "owin" (for observation window ) represents a spatial region or window in the two-dimensional plane. A window usually represents our study area : the window of observation of a point pattern, or the region where we want to make predictions, etc. To create a window object we can build one from data in R, using owin and other tools; extract the window from one of the point pattern datasets supplied with the package by typing W <- X$window where X is the point pattern; convert data from other formats using as.owin; manipulate existing windows using a wide variety of tools or derive a window from a point pattern or pixel image using various tools.

11 Journal of Statistical Software 11 Figure 8: Polygonal window (left) and pixellated window (right). The shape of a window is almost arbitrary; it may be a rectangle, a polygon, a collection of polygons (including holes), or a binary image mask. See Figure 8. spatstat supports polygonal windows of arbitrary shape and topology. That is, the boundary of the window may consist of one or more closed polygonal curves, which do not intersect themselves or each other. The window may have holes. spatstat also supports pixellated windows. A matrix with logical entries is interpreted as a binary pixel image whose entries are TRUE where the corresponding pixel belongs to the window. Pixellated windows can be created from raw data, read from data files, or created by analytic equations. They are also produced in spatstat by various geometrical operations, such as morphological erosion Pixel images An object of the class "im" represents a pixel image. It is essentially a matrix of numerical values associated with a rectangular grid of points inside a window in the x, y plane. A pixel image may be displayed on the screen as a digital image, a contour map, or a relief surface. Image objects can be created explicitly using im. Data in other formats can be converted to an "im" object using as.im Figure 9: Example of pixel image data. Top: line segment pattern from the copper dataset. Bottom: a pixel image derived from the copper data. Pixel value is the distance to the nearest line segment. A pixel image may contain real experimental data, for example, a satellite image of the study

12 12 spatstat: An R Package for Analyzing Spatial Point Patterns region. One of the important roles of pixel images is to provide covariate data for statistical models. The brightness value of the image at a particular pixel is the value of the spatial covariate at that location. For example, Figure 9 shows a colour image derived from the spatial covariates in the copper dataset. Figure 10: A computed pixel image (displayed as a contour plot): the distance transform of a point pattern. Obtained by contour(distmap(x)) where X was the point pattern. Dots indicate original point pattern dataset. Pixel images are also produced by many functions in spatstat, for example when we apply kernel smoothing to point pattern data (ksmooth.ppp), when we estimate the second moment measure of a point process (Kmeasure), compute the geometric covariance of a window (setcov) or evaluate the distance map of a point pattern (distmap). See Figure 10. We also use pixel images to represent mathematical functions of the Cartesian coordinates. Any function object f(x,y) in R can be converted into a pixel image using as.im. 6. Operations on data Once we have created a point pattern dataset, it can be inspected, plotted and modified using the commands described here Basic inspection of data There are print, summary and plot methods for point patterns, windows, and pixel images. R> hamster marked planar point pattern: 303 points multitype, with levels = dividing, pyknotic Window: rectangle = [ 0, 1 ] x [ 0, 1 ] R> summary(hamster)

13 Journal of Statistical Software 13 Marked planar point pattern: 303 points Average intensity 303 points per unit area Marks: frequency proportion intensity dividing pyknotic Window: rectangle = [0,1] x [0,1] Window area = 1 R> plot(hamster) Plotting is isometric, i.e. the physical scales of the x and y axes are the same. For marked point patterns, the plotting behaviour depends on whether the marks are continuous or categorical, and typical displays are shown in Figures 6 and 7 respectively. To see the locations of the points without the marks, type plot(unmark(x)). The colours, plotting characters, line widths and so on can be modified by adding arguments to the plot methods. Default plotting behaviour can also be controlled using the function spatstat.options. The function identify.ppp, a method for identify, allows the user to examine a point pattern interactively Subsets of point patterns spatstat supports the extraction of subsets of a point pattern, with a method for the indexing operator "[". This performs either thinning (retaining/deleting some points of a point pattern) or trimming (reducing the window of observation to a smaller subregion and retaining only those points which lie in the subregion). If X is a point pattern object then X[subset, ] will cause the point pattern to be thinned, retaining only the points indicated by subset. The latter can be any type of subset argument such as a positive integer vector, a logical vector, or a negative integer vector (the latter indicating which points should be deleted). The pattern will be trimmed if we call X[, window] where window is an object of class "owin". Only those points of X lying inside the new window will be retained Other operations on point patterns Marks can readily be added to and removed from a point pattern using the functions unmark and setmarks or the operator %mark%. Marks can be manipulated rapidly using the methods for cut, split and split<- for point patterns. For a point pattern with numerical marks, cut.ppp will transform the marks into factor levels. For a multitype point pattern, split.ppp will separate the dataset into a list of point patterns, each consisting of points of one type. The functions superimpose and "split<-.ppp" will combine several point patterns into a single point pattern, attaching mark labels if required. Geometrical operations on point patterns include planar rotation, translation and affine transformation (rotate, shift and affine). There are functions to compute the distance

14 14 spatstat: An R Package for Analyzing Spatial Point Patterns from each point to its nearest neighbour (nndist), the distance between each pair of points (pairdist) and the distance from each point to the boundary of the window (bdist.points) Manipulating windows The following functions are available for manipulating windows. bounding.box Find smallest rectangle enclosing the window with sides parallel to the x and y axes erode.owin Erode window by a distance r rotate.owin Rotate the window shift.owin Apply vector translation affine.owin Apply an affine transformation complement.owin Invert (inside outside) is.subset.owin Test whether one window contains another trim.owin Intersect window with rectangle intersect.owin Intersection of windows union.owin Union of windows ripras Estimate window from points Pixellating windows The shape of any spatial region may be approximated by a binary pixel image. In spatstat the image is represented as a window object (class "owin") of type "mask". The following commands are useful. as.mask Convert to pixel approximation raster.x Extract the x coordinates of the pixel raster raster.y Extract the y coordinates of the pixel raster The default accuracy of the approximation can be controlled using spatstat.options. Additionally nearest.raster.point maps continuous cartesian coordinates to raster locations. Geometrical computations with windows The following commands are useful for computing geometrical quantities. inside.owin area.owin diameter eroded.areas bdist.points bdist.pixels centroid.owin distmap Test whether (x, y) points are inside window Compute window s area Compute window s diameter Compute areas of eroded windows Compute distances from data points to window boundary Compute distances from all pixels to window boundary Compute centroid (centre of mass) Compute distance transform of window 6.5. Pixel images Functions which return a pixel image include the following.

15 Journal of Statistical Software 15 Kmeasure setcov ksmooth.ppp distmap Reduced second moment measure of point pattern Set covariance function of spatial window Kernel smoothed intensity estimate of point pattern Distance transform of point pattern Functions which manipulate a pixel image include the following. im as.im plot.im contour.im persp.im [.im shift.im print.im summary.im is.im 6.6. Programming tools Create a pixel image Convert data to pixel image Display as digital image Display as contour map Display as perspective view Extract subset of pixel image Apply vector shift to pixel image Print basic information Print summary Test whether object is a pixel image spatstat also contains some programming tools to assist in calculations with point patterns. One of these is the function applynbd which can be used to visit each point of the point pattern, identify its neighbouring points, and apply any desired operation to these neighbours. For example the following code calculates the distance from each point in the pattern redwood to its second nearest neighbour: R> nnd2 <- applynbd(redwood, N = 2, exclude=true, function(y, cur, d, r){max(d)}) This has obvious applications for LISA methods Anselin (1995); Cressie and Collins (2001b,a). One can also use applynbd to perform animations in which each point of the point pattern is visited and a graphical display is executed. There is an example in demo(spatstat). 7. Exploratory data analysis The literature on spatial statistics contains a very large number of techniques for the exploratory analysis of point pattern data. Perhaps the most famous example is Ripley s K- function. As far as we know, the vast majority of these techniques have never been implemented in public domain software, apart from the initial proof-of-concept implementations by their original authors. The uptake of new methods in practice seems to have been severely limited by the lack of such software. Accordingly, one of the main aims of the spatstat project is to implement the existing, published techniques of spatial statistics in open source software Initial inspection of data Initial, interactive inspection of a point pattern dataset is supported by the methods for print, summary, plot and identify mentioned above. The function summary.ppp computes the average intensity of points, summarises the marks if X is a marked point pattern, and

16 16 spatstat: An R Package for Analyzing Spatial Point Patterns describes the window. Subsets of the data can be extracted using the methods for "[", cut and split Spatial inhomogeneity One of the important questions about a point pattern dataset is whether it can be treated as spatially homogeneous. To investigate this, Diggle and others have recommended kernel smoothing. The function ksmooth.ppp performs kernel smoothing of a point pattern, and yields a pixel image object. 0 1e 06 2e 06 3e 06 4e 06 5e 06 6e 06 Figure 11: Kernel smoothed intensity estimate for the point pattern in Figure 2, indicating a clear trend from left to right. spatstat contains several functions which extend classical techniques (developed for homogeneous patterns) to inhomogeneous point patterns. They include Kinhom (an inhomogeneous version of the K function Baddeley et al. (2000)) and the model-fitting function ppm Summary statistics for unmarked point patterns Exploratory analysis of point patterns is based largely on summary statistics. The spatstat package will compute estimates of the summary functions F (r), the empty space function (contact distribution or point-to-event distribution) G(r), the nearest neighbour distance distribution function ( event-to-event distribution) J(r), the function J = (1 G)/(1 F ) K(r), the reduced second moment function ( Ripley s K function ) g(r), the pair correlation function g(r) = [ d dr K(r)]/(2πr) for a point pattern, and their analogues for marked point patterns. These estimates can be used for exploratory data analysis and in formal inference about a spatial point pattern. They are well described in the literature, e.g. Ripley (1981), Diggle

17 Journal of Statistical Software 17 (2003), Cressie (1991), (Stoyan et al. 1995, Chapter 15), Stoyan and Stoyan (1995). The J-function was introduced in van Lieshout and Baddeley (1996). The point pattern is assumed to be stationary (homogeneous under translations) in order that the functions F, G, J, K be well-defined and the corresponding estimators approximately unbiased. (There is an extension of the K function to inhomogeneous patterns; see below). The corresponding spatstat library functions are: Fest estimate of empty space function F Gest estimate of nearest neighbour distribution function G Jest estimate of J-function Kest estimate of Ripley s K-function allstats estimates of all four functions F, G, J, K pcf estimate of pair correlation function g (Some others are listed below). In each of these commands, the user has a choice of several alternative estimation methods. These estimators are based on different edge corrections, or strategies for removing the bias due to edge effects, which arise because we only observe the point pattern inside a restricted spatial window. Several dozen alternative edge corrections have been published in the literature; see Baddeley (1998); Stoyan and Stoyan (1995) for surveys. Part of the spatstat project is to implement all of these proposed estimators so that they may be compared in practice. The routines Fest, Gest, Jest, Kest, pcf each return an object of class "fv" (for function value ). This is a data frame with some extra attributes indicating the recommended way of plotting the function, and other information. It is a convenient way of storing (particularly for use in future plotting) several different estimates of the same function. A column labelled r in this data frame contains the values of the argument r for which the summary function ( F (r), etc) has been evaluated. Other columns give the estimates of the summary function itself, using several competing estimators. Along with the different function estimates, the data frame includes the vector of theoretical expected values (theo) that the function would have under the assumption of complete spatial randomness (CSR) i.e. under a homogeneous Poisson point process model. There are methods for print and plot for the class "fv". The plot method is particularly useful. It is a generalisation of plot.formula, and enables the summary functions to be re-plotted in a variety of ways. There are various recommendations in the literature about how to plot the summary functions to reveal diagnostic information. An aim of spatstat is to make it easy to plot the summary functions in different ways. Probably the most common exploratory graphic is a plot of K(r) against r. An example of a useful transformed graphic is a plot of L(r) = K(r)/π against r, as recommended by Ripley (1981), the rationale being that this procedure linearizes the plot and stabilizes the variance. Diggle (1983, 2003) recommends plotting K(r) πr 2 against r, so as to remove the mean. These plots can be achieved as follows: R> Kc <- Kest(cells) R> plot(kc)

18 18 spatstat: An R Package for Analyzing Spatial Point Patterns K(r) data Poisson r Figure 12: Output of plot(kest(x)). R> plot(kc, cbind(r, sqrt(iso/pi)) ~ r) R> plot(kc, cbind(trans,iso,border) - theo ~ r) Notice the use of cbind in the last two plots. The effect is that several functions (the columns in the cbind expression on the left hand side) will be plotted in the same plot, against the variable on the right hand side of the formula. With respect to the empty space (contact) distribution function F, Ripley (1981, 1988) simply plots F (r) against r, whereas Diggle (2003) plots F (r) against F 0 (r) = 1 exp{ ˆλπr 2 }, this being the form of F under the assumption of complete spatial randomness. This is in effect a P P plot. Another useful graphic (suggested by Murray Aitkin) is a plot of sin 1 ( F (r)) against sin 1 ( F 0 (r)). The function g(x) = sin 1 x is Fisher s variancestabilising transformation for the binomial estimator of a proportion, and indeed seems to approximately stabilise the variance in this context. These alternative plots may be displayed as follows. R> Fc <- Fest(cells) R> plot(fc) R> plot(fc, cbind(km, trans, border) ~ theo) R> fisher <- function(x) { asin(sqrt(x)) } R> plot(fc, fisher(cbind(km, trans, border)) ~ fisher(theo)) Initially it may be unclear which of the summary functions will provide insight, and it is usually desirable to calculate and plot estimates of all four. The command plot(allstats(x)) will produce a plot of estimates of the four main summary functions K, F, G and J. Distances between points are also computed (without edge correction) by:

19 Journal of Statistical Software 19 nndist pairdist exactdt nearest neighbour distances distances between all pairs of points distance from any location to nearest data point There are also several related alternative functions. For the second order statistics, alternatives are: Kinhom K function for inhomogeneous point patterns Kest.fft fast K-function using FFT for large datasets Kmeasure reduced second moment measure The function Kmeasure yields a pixel image of the estimated Reduced Second Moment Measure K (see (Stoyan and Stoyan 1995, p. 245)). This measure is the Fourier transform of the Bartlett spectrum Bartlett (1964, 1975). Although first defined in the 1960 s this concept appears not to have been implemented in software until recently. Its usefulness in data analysis is yet to be explored. 0 5e 05 1e e 04 Figure 13: The cells dataset (Left) and a density estimate of its second moment measure (Right). Figure 13 shows the well-known cells dataset, and a density estimate of its second moment measure, computed by Kmeasure. [The algorithm takes the raw Bartlett periodogram, multiplies by the Fourier transform of the bivariate normal density, then takes the inverse FFT to yield the smoothed density.] The large contour in the centre of the Figure is a region of low second moment density close to the origin, caused by the spatial inhibition between points at short distances. The pronounced non-circular shape of this contour suggests that the interpoint interaction is anisotropic, which does not appear to have been noticed before. 8. Summary statistics for multitype point patterns Analogues of the G, J and K functions have been defined in the literature for multitype point patterns, that is, patterns in which each point is classified as belonging to one of a finite number of possible types (e.g. on/off, species, colour). The best known of these is the bivariate (cross) K function K ij (r) derived by counting, for each point of type i, the number

20 20 spatstat: An R Package for Analyzing Spatial Point Patterns of type j points lying closer than r units away. The corresponding nearest-neighbour function G ij (r) is the distribution of the distance from a typical point of type i to the nearest point of type j. Using the symbol to denote points of any type (i.e. all points regardless of their type) we may define analogous functions K i and G i. For further explanation see van Lieshout and Baddeley (1999). Gcross,Gdot,Gmulti multitype nearest neighbour distributions G ij, G i Kcross,Kdot, Kmulti multitype K-functions K ij, K i Jcross,Jdot,Jmulti multitype J-functions J ij, J i These functions operate in a very similar way to Gest, Jest, Kest with additional arguments specifying the type(s) of points to be studied Function arrays For multitype patterns we might want to compute a summary function for the points of type i for each of the possible types of the pattern. Alternatively we might want to compute a summary function for each possible pair of types (i, j). A function array is a collection of functions f i,j (r) indexed by integers i and j. An example is the set of cross K functions K ij (r) for all possible pairs of types i and j in a multitype point pattern (1 i, j m where m is the number of types). It is best to think of this as a genuine matrix or array. A function array is represented in spatstat by an object of type "fasp" (function array for spatial patterns). It can be stored, plotted, indexed and subsetted in a natural way. If Z is a function array, then R> plot(z) R> plot(z[,3:5]) will plot the entire array, and then plot the subarray consisting only of columns 3 to 5. The function alltypes will compute a summary statistic for each possible type, or each possible pair of types, in a multitype point pattern. The value returned by alltypes is a function array object. For example if X is a multitype point pattern with 3 possible types, R> Z <- alltypes(x, "K") yields a 3 3 function array such that (say) Z[1,2] represents the cross-type K function K 1,2 (r) between types 1 and 2. The command plot(z) will then plot the entire set of cross K functions as a two-dimensional array of plot panels. Arguments to plot.fasp can be used to change the plotting style, the range of the axes, and to select which estimator of K ij is plotted. These options apply to all the plot panels simultaneously. The command allstats yields a 2 2 function array containing the F, G, J and K functions of an (unmarked) point pattern.

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